HOFER GEOMETRY AND COTANGENT FIBERSmath.uga.edu/~usher/cotangent.pdf · HOFER GEOMETRY AND COTANGENT FIBERS 3 denote the orbit of S under the Hamiltonian diffeomorphism group. The

HOFER GEOMETRY AND COTANGENT FIBERS

MICHAEL USHER

ABSTRACT. For a class of Riemannian manifolds that include products of arbitrary compact mani-folds with manifolds of nonpositive sectional curvature on the one hand, or with certain positive-curvature examples such as spheres of dimension at least 3 and compact semisimple Lie groupson the other, we show that the Hamiltonian diffeomorphism group of the cotangent bundle con-tains as subgroups infinite-dimensional normed vector spaces that are bi-Lipschitz embedded withrespect to Hofer’s metric; moreover these subgroups can be taken to consist of diffeomorphismssupported in an arbitrary neighborhood of the zero section. In fact, the orbit of a fiber of the cotan-gent bundle with respect to any of these subgroups is quasi-isometrically embedded with respect tothe induced Hofer metric on the orbit of the fiber under the whole group. The diffeomorphisms inthese subgroups are obtained from reparametrizations of the geodesic flow. Our proofs involve astudy of the Hamiltonian-perturbed Floer complex of a pair of cotangent fibers (or, more generally,of a conormal bundle together with a cotangent fiber). Although the homology of this complexvanishes, an analysis of its boundary depth yields the lower bounds on the Lagrangian Hofer metricrequired for our main results.

1. INTRODUCTION

For a symplectic manifold (P,ω), let Ham(P,ω) denote the group of Hamiltonian diffeo-morphisms φ of P which may be obtained as time-one maps of compactly-supported smoothfunctions H : [0, 1]× P → R. (Thus where XH(t, ·) is the time-dependent vector field given by

ω(·, XH(t, ·)) = d(H(t, ·)) and where φ tH0≤t≤1 is defined by φ0

H= 1P and

dφ tH

d t= XH(t,·) φ t

H

we have φ1H= φ). According to [Ho90],[LM95], the “Hofer norm”

‖φ‖= inf

(∫ 1

0

max

PH(t, ·)−min

PH(t, ·)

d t

φ1H= φ

)

gives rise via the formula d(φ,ψ) = ‖φ ψ−1‖ to a bi-invariant metric d on Ham(P,ω).Despite substantial progress, much remains unknown about the large-scale properties of this

metric. In particular, it is still unknown even whether d is unbounded for every symplectic man-ifold, though unboundedness has now been established in many cases (see, e.g., [Mc09],[U11b]and references therein).

In the present note we concentrate on the case where the sympelctic manifold (P,ω) is thecotangent bundle (T ∗N , dθ ) of a compact smooth manifold N , with its standard symplecticstructure. In this case the unboundedness of the Hofer metric has been known at least since[M01], but many more refined questions remain. We investigate some of these here by consid-ering the geodesic flow on N with respect to a suitable Riemannian metric; in particular, undera Morse-theoretic assumption on the geodesics in N , we will exhibit infinite-rank subgroups ofHam(T ∗N , dθ ) which are “large” with respect to Hofer’s metric.

To set up terminology for one of our main results, if (N , g) is a compact Riemannian manifold,Q ⊂ N is a compact submanifold, and x1 ∈ N\Q, a geodesic from Q to x1 is by definition a solutionη: [0, 1]→ N to the geodesic equation∇η′η′ = 0 subject to the boundary conditions η(0) ∈Q,

1

2 MICHAEL USHER

η′(0) ⊥ Tη(0)Q, and η(1) = x1. Letting PN (Q, x1) denote the space of all piecewise-smoothpaths η: [0, 1]→ N such that η(0) ∈Q and η(1) = x1, the geodesics from Q to x1 are precisely

the critical points of the energy functional E : PN (Q, x1)→ R defined by E(η) =∫ 1

0|η′(t)|2d t.

In particular, any geodesic from Q to x1 has a well-defined Morse index, which will be referredto in Theorem 1.1 below.

Also, let R∞ denote the direct sum of a collection of copies of R indexed by Z+, and for~a = ak∞k=1 ∈ R∞ define osc(~a) = maxi, j |ai − a j | and ‖~a‖∞ = maxi |ai | (these maxima arewell-defined since, by definition, any ~a ∈ R∞ has all but finitely many ai = 0). Obviously onehas ‖~a‖∞ ≤ osc(~a)≤ 2‖~a‖∞. We prove:

Theorem 1.1. Let (N , g) be a compact connected Riemannian manifold and suppose that there is

a compact submanifold Q ⊂ N, a point x1 ∈ N \Q which is not a focal point of Q, and a homotopy

class c ∈ π0(PN (Q, x1)) such that:

• No geodesics from Q to x1 representing the class c have Morse index one.

• Only finitely many geodesics from Q to x1 representing the class c have Morse index in

0, 2.• dimQ 6= dim N − 2.

Then for any neighborhood U of the zero section 0N ⊂ T ∗N there is a homomorphism Φ : R∞ →Ham(T ∗N , dθ ) such that every diffeomorphism Φ(~a) has support contained in U \ 0N and such

that, for all ~a,~b ∈ R∞

(1) ‖~a− ~b‖∞ ≤ d(Φ(~a),Φ(~b))≤ osc(~a− ~b)

Thus for all manifolds N admitting metrics g, submanifolds Q, and points x1 as in Theorem1.1, the Hamiltonian diffeomorphism group of an arbitrarily small deleted neighborhood U \0N

of the zero section contains as a subgroup an infinite-dimensional normed vector space whichis bi-Lipschitz embedded in Ham(T ∗N , dθ ) with respect to the Hofer metric. The hypothesesof Theorem 1.1 can at least formally be weakened somewhat (see Assumption 3.4 below; thehypotheses of Theorem 1.1 amount to Assumption 3.4 being satisfied with k = 0), though I donot know any examples of manifolds that satisfy Assumption 3.4 for some nonzero k but do notalso satisfy it for k = 0. Let us point out here some examples of classes of manifolds to whichTheorem 1.1 applies; see the discussion following Assumption 3.4 for proofs and additionalremarks:

• Any compact Riemannian manifold (N , g) of nonpositive sectional curvature (see Propo-sition 3.9). Indeed, if dim N 6= 2 we can take Q to consist of an arbitrary single pointdistinct from x1, while if dim N = 2 we can take Q to be a closed geodesic not containingx1.• Many positively-curved symmetric spaces, including spheres of dimension at least 3,

all compact semisimple Lie groups, and quaternionic Grassmannians (see Proposition3.10 and Remark 3.11).• All products N × N ′, where N is a compact smooth manifold admitting a Riemannian

metric g, submanifold Q, and point x1 as in the hypothesis of Theorem 1.1, and whereN ′ is any compact smooth manifold (see Remark 3.12).

In the course of proving Theorem 1.1, we establish a result concerning the behavior of theHofer metric with respect to Lagrangian submanifolds of T ∗N that is also of interest. In general,if (P,ω) is a symplectic manifold and if S ⊂ P is a closed subset, let

L (S) = φ(S)|φ ∈ Ham(P,ω)

HOFER GEOMETRY AND COTANGENT FIBERS 3

denote the orbit of S under the Hamiltonian diffeomorphism group. The Hofer norm ‖ · ‖ onHam(P,ω) induces a Ham(P,ω)-invariant pseudometric δ on L (S), via the formula

δ(S0, S1) = inf‖φ‖|φ ∈ Ham(P,ω), φ(S0) = S1.Here we will consider the case where (P,ω) = (T ∗N , dθ ) and S is the fiber T ∗

x1N over a point

x1 ∈ N or, more generally, the conormal bundle

ν∗Q =(x , p) ∈ T ∗N |Qp|TxQ = 0

of a submanifold Q ⊂ N .

Theorem 1.2. Let (N , g), Q ⊂ N, and x1 ∈ N \Q be as in Theorem 1.1, and let U ⊂ T ∗N be a

neighborhood of the zero section 0N . Then there is a linear map

F : R∞→ C∞(T ∗N)

such that each F(~a) has compact support contained in U \ 0N , such that for all ~a ∈ R∞

(2) max F(~a)−min F(~a) = osc(~a),

and such that, for some constant C > 0 and for all ~a,~b ∈ R∞ we have

(3) φ1F(~a) φ1

F(~b)= φ1

F(~a+~b),

(4) δφ1

F(~a)(T∗x1

N),φ1F(~b)(T ∗

x1N)

≥ ‖~a− ~b‖∞ − C

and

(5) δφ1

F(~a)(ν∗Q),φ1

F(~b)(ν∗Q)≥ ‖~a− ~b‖∞ − C .

In view of (2), (3), and the Ham-invariance of δ we also have

δφ1

F(~a)(T∗x1

N),φ1F(~b)(T ∗

x1N)

≤ ‖φ1

F(~b−~a)‖ ≤ osc(~a− ~b)

and likewise with ν∗Q in place of T ∗x1

N . Thus Theorem 1.2 gives quasi-isometric embeddings of

R∞ into the metric1 spaces (L (T ∗x1

N),δ) and (L (ν∗Q),δ).Theorem 1.2 quickly implies Theorem 1.1, as we now show:

Proof of Theorem 1.1, assuming Theorem 1.2. Given F : R∞→ C∞(T ∗N) as in Theorem 1.2 de-fine Φ : R∞→ Ham(T ∗N , dθ ) by Φ(~a) = φ1

F(~a). By (3), Φ is a homomorphism. Moreover

d(Φ(~a),Φ(~b)) = ‖φ1F(~a−~b)‖ ≤ osc(~a− ~b)

by (3) and (2), proving the upper bound in (1).As for the lower bound, since Φ is a homomorphism we just need to show that, for all

~a ∈ R∞, ‖Φ(~a)‖ ≥ ‖~a‖∞. Suppose that this is false, and let ε > 0 be such that ‖Φ(~a)‖ ≤‖~a‖∞ − ε. Where C is the constant in Theorem 1.2, choose m ∈ Z+ such that mε > C .Then since Φ is a homomorphism and since the Hofer norm satisfies the triangle inequalitywe would have ‖Φ(m~a)‖ ≤ m(‖~a‖∞ − ε) < ‖m~a‖∞ − C . But this contradicts the fact that, by(4), δ(Φ(m~a)T ∗

x1N , T ∗

x1N)≥ ‖m~a‖∞ − C .

1The fact that δ defines a metric, and not just a pseudometric, onL (T ∗x1N) andL (ν∗Q) follows from [U12, Remark

4.12].

4 MICHAEL USHER

1.1. Related work. Theorem 1.2 is the first result known to the author about the Hofer ge-ometry of the orbit space L (T ∗

x1N) of a cotangent fiber under the Hamiltonian diffeomorphism

group. However there are several prior results about the Hofer metric on Ham(T ∗N , dθ ) whichimply conclusions closely related to those of Theorem 1.1, especially in the case that (N , g) hasnonpositive sectional curvature.

First of all, [M01] contains results about the orbitL (0N ) of the zero section of a the cotangentbundle of an arbitrary compact smooth manifold, nicely complementing our results about theorbits of fibers or of certain conormal bundles. Namely, [M01, Proposition 1(5,6)] shows that,for any compact N , the map C∞(N)→ L (0N ) which assigns to a smooth function S : N → Rthe Lagrangian submanifold graph(dS) induces an isometric embedding ( C∞(N)

R, osc)→L (0N ),

where we write osc([ f ]) =max f −min f . Using this, one easily obtains isometric embeddingsthat map arbitrarily large balls in the normed vector space (C∞(N)/R, osc) into Ham(T ∗N , dθ ).However, in view of the fact that Ham(T ∗N , dθ ) contains only compactly supported diffeomor-phisms, it does not seem possible to obtain embeddings of a whole vector space using such aconstruction; moreover, in order to embed large balls one must use diffeomorphisms with largesupports, in contrast to Theorem 1.1 where the supports are fixed.

In [Py08], Py showed that, whenever a symplectic manifold (P,ω) contains a π1-injectivecompact Lagrangian submanifold L which admits a Riemannian metric of nonpositive sectionalcurvature, for any integer k there exists a bi-Lipschitz embedding of Rk into Ham(M ,ω) (withthe relevant constants diverging to ∞ with k), whose image moreover consists of diffeomor-phisms which may be taken to have support in an arbitrary deleted neighborhood of L. Thusin the case that P = T ∗N and L is the zero section, Theorem 1.1 represents an improvement onPy’s result both in that the embedded subgroups are infinite-dimensional and in that the classof manifolds N to which the theorem applies is more general than just those of nonpositive cur-vature. In the case that P is instead compact, a similar improvement of Py’s result is a specialcase of [U11b, Theorem 1.1].

More recently, in [MVZ12, Theorem 1.13] it was shown that if a compact manifold N ad-mits a nonsingular closed one-form then Ham(T ∗N , dθ ) admits an isometric embedding of(C∞

c((0, 1)), osc).

Finally, L. Polterovich has pointed out to the author a way of getting a conclusion closelyrelated to that of Theorem 1.1 in the special case that (N , g) has nonpositive curvature (forsimilar arguments in the contact context see [FPR12, Examples 2.8-10]). Namely, if for r > 0we let S(r)∗N denote the radius-r cosphere bundle, one can see from [Gi07, Theorem 2.7(iii)]that S(r)∗N is stably nondisplaceable (since due to the curvature hypothesis N has no con-tractible closed geodesics). Then arguing as in [Po01, Proposition 7.1.A] one can show thatfor any compactly supported smooth function H : T ∗N → R, the Hofer (pseudo-)norm ‖φH‖ ofthe corresponding element φH in the universal coveràHam(T ∗N , dθ ) is greater than or equal tomaxS(r)∗N |H|. One can then let r vary and consider Hamiltonians of the form H f = f ρ whereρ is the fiberwise norm on T ∗N and where f : (0,∞) → R is a compactly-supported smoothfunction. The assignment f 7→ φH f

is then an embedding C∞c((0,∞)) ,→àHam(T ∗N , dθ ) which

obeys an estimate analogous to (1). In general, it does not seem clear from this argumentwhether the lower bound in this estimate survives after one projects down fromàHam(T ∗N , dθ )

to Ham(T ∗N , dθ ), though in some isolated cases such as where N is the torus this could likelybe deduced along the lines of [FPR12, Example 2.8]. Also, we should mention that the diffeo-morphisms that appear in the image of our embedding Φ in Theorem 1.1 have the form φH f

forcertain functions f ; thus the proof of Theorem 1.1 will show that Polterovich’s estimate in theuniversal cover does survive after projection at least for these special choices of f .


1.2. Summary of the proof. Our proof of Theorem 1.2 is based on the properties of the filtra-tions of the Floer complexes associated to certain pairs of noncompact Lagrangian submanifoldsin Liouville manifolds; in our case the Liouville manifold is a cotangent bundle T ∗N and theLagrangian submanifolds are cotangent fibers T ∗

x1N or, a bit more generally, conormal bundles

ν∗Q of compact submanifolds Q ⊂ N . Thus if x1 /∈Q, for any suitably nondegenerate, compactlysupported Hamiltonian H : [0, 1]×T ∗N → Rwe have a Floer complex C F(ν∗Q, T ∗

x1N ; H)whose

generators correspond to points of ν∗Q ∩ (φ1H)−1(T ∗

x1N).

Since we are considering compactly supported Hamiltonians (unlike the situation with the“wrapped” Floer complexes of, e.g., [AS10]), the homology of this Floer complex is trivial, asν∗Q ∩ T ∗

x1N = ∅. Despite this triviality, we can still obtain significant information (depending,

unlike the homology, on the Hamiltonian H) from the Floer complex, by means of the boundary

depth, a quantity that was introduced in a similar context in [U11a] and is denoted here byB(ν∗Q, (φ1

H)−1(T ∗

x1N)). The boundary depth gives a quantitative measurement of the nontrivi-

ality of the Floer boundary operator with respect to the natural filtration on the Floer complex:it is the smallest nonnegative number β such that every element c of the image of the boundaryoperator has a primitive whose filtration level is at most β larger than that of c. As we show inSection 2 (by arguments quite analogous to ones appearing in [U11b, Section 6] in the compactcase), the boundary depth B(ν∗Q,Λ) is independent of the choice of H with (φ1

H)−1(T ∗

x1N) = Λ

and moreover, considered as a function of Λ, is 1-Lipschitz with respect to the Hofer distance δon L (T ∗

x1N). Consequently lower bounds for the boundary depth can give lower bounds on δ

of the sort that appear in (4).Thus we prove Theorem 1.2 by constructing a homomorphism

R∞→ Ham(T ∗N , dθ )

~a 7→ φ1F(~a)

where osc(F(~a)) = osc(~a) for all ~a while B(ν∗Q, (φ1F(~a))−1(T ∗

x1N))≥ ‖a‖∞− C for a constant C .

For F(~a) we use a function of the form f~a ρ, where ρ : T ∗N → R is the fiberwise norm givenby our Riemannian metric g on N and f~a : [0,∞)→ R is a function whose graph is as in Figure1.2, with the heights of its minima and maxima given by the coordinates ai of ~a. (More detailon the functions f~a is given at the start of Section 4; our argument requires them to take a ratherspecial form, though this constraint is likely only technical.) Recall that the function ρ2 on T ∗Nhas Hamiltonian flow given by the geodesic flow of g; consequently f~a ρ will restrict to eachcosphere bundle S(r)∗N as the time-one map of a reparametrization of the geodesic flow. Apoint (x , p) will lie in ν∗Q∩ (φ1

F(~a))−1(T ∗

x1N) if and only if x ∈Q, p ∈ T ∗

xN vanishes on TxQ, and

±p is the initial momentum of a geodesic of length ± f ′~a(|p|) which begins at x and ends at x1.Thus a geodesic η in N which starts orthogonally to Q and ends at x1 gives rise to several

generators for the Floer complex C F(ν∗Q, T ∗x1

N ; F(~a)), one for each value of r such that | f ′~a(r)|

is equal to the length of η. In order for two generators of the complex to be intertwined by theFloer boundary operator, their corresponding geodesics must be homotopic (through paths fromQ to x1), and their Floer-theoretic gradings must differ by 1. We compute the Floer-theoreticgrading in Proposition 3.3; this grading is found to depend in an explicit way on the Morseindex of the geodesic η and on the signs of both f ′

~a(r) and f ′′

~a(r). Meanwhile the action of the

generator is approximately f~a(r).Provided that (−mini ai) is sufficiently large, there will be a generator c0 of C F(ν∗Q, T ∗

x1N ; F(~a))

whose associated geodesic from Q to x1 has index zero and whose associated value of r obeysf ′~a(r), f ′′

~a(r) > 0, such that the action of c0 is approximately mini ai . Under Assumption 3.4

6 MICHAEL USHER

FIGURE 1. The graph of a typical function f~a. In this case ~a =

(4,−6, 1, 2,−4, 0, 0, . . .) ∈ R∞, with the coordinates of ~a equal to the heights ofthe extrema of f~a as one moves from right to left.

(as applies in Theorem 1.2), this generator will be a cycle in the Floer complex. So sincethe Floer homology vanishes, c0 must also be a boundary. But Assumption 3.4 also impliesthat every generator whose exceeds that of c0 by 1 has the property that its associated valueof r obeys f ′′

~a(r) < 0, with | f ′

~a(r)| bounded above independently of ~a. The special form of

our functions f~a then implies that f~a(r) is bounded below independently of ~a. Thus all possi-ble primitives of c0 have action bounded below by a constant, whereas c0 has action approxi-mately mini(ai). This leads to an estimate B(ν∗Q, (φ1

F(~a))−1(T ∗

x1N)) ≥ −mini(ai) − C . A sim-

ilar argument, together with a duality property satisfied by the boundary depth, implies thatB(ν∗Q, (φ1

F(~a))−1(T ∗

x1N)) ≥ maxi(ai) − C . Combining these two estimates quickly implies (4).

Another duality argument yields the similar estimate (5) and hence completes the proof of The-orem 1.2.

We remark that for many cases of Theorem 1.2 it would suffice to study Floer complexes of theform C F(T ∗

x0N , T ∗

x1N ; H) (i.e., to set Q equal to a single point). However, when dim N−dimQ =

2 one finds from the grading formula in Proposition 3.3 that the argument indicated in theprevious paragraph breaks down, and so in particular one cannot set Q equal to a singletonin the case that N is two-dimensional. So when N is a surface of nonpositive curvature weinstead set Q equal to a closed geodesic on N that does not contain x1. Of course, incorporatingpositive-dimensional Q into our discussion also allows us to obtain the more general results onthe Hofer geometry ofL (ν∗Q)mentioned in (5); furthermore, allowing positive-dimensional Q

in the hypothesis of Theorem 1.1 leads to the hypothesis being preserved under products witharbitrary compact manifolds, as noted in Remark 3.12.

1.3. Outline of the paper. In the following Section 2 we set up the general framework forthe Floer complex associated to a Hamiltonian function and a pair of suitable noncompact La-grangian submanifolds of a Liouville manifold, and we define the boundary depth associated tothis complex and prove the basic result Corollary 2.4 connecting the boundary depth to Hofergeometry. Section 3 specializes the discussion to the case in which the Liouville manifold is thecotangent bundle T ∗N of a Riemannian manifold, the Hamiltonian is a function of the normgiven by the Riemannian metric, and the pair of the Lagrangian submanifolds consists of a


conormal bundle ν∗Q and a cotangent fiber T ∗x1

N . In this case the Floer complex can be un-derstood in terms of geodesics from Q to x1. For our main results it is important to work outthe grading on the complex, as we do in detail in Section 3.1; this is somewhat easier whenQ consists of a single point (which as mentioned earlier suffices for many cases of Theorem1.2), so we treat that case initially, followed by a discussion of the necessary modifications inthe more general case. Next, in Section 3.2, we introduce Assumption 3.4 on the behavior ofgeodesics connecting Q to x1 and discuss some situations in which it holds. This assumption isa somewhat broader version of the hypotheses of Theorems 1.1 and 1.2, and is the most generalcontext in which we are able to prove such results.

Finally, in Section 4, we introduce a special class of Hamiltonians for which the gradingcomputation of Section 3.1 allows us to get strong lower bounds on the boundary depth un-der Assumption 3.4, and we prove these lower bounds. From these lower bounds we deduceCorollary 4.4, which has Theorem 1.2 as a special case, thus completing the proofs of our mainresults.

Acknowledgements. I am grateful to Leonid Polterovich for asking me the question whichmotivated this work and for various helpful comments. The work was supported by NSF grantDMS-1105700.

2. GENERALITIES ON THE FLOER COMPLEX AND THE BOUNDARY DEPTH

Let (M ,θ ) be a Liouville domain; thus M is a compact manifold with boundary, with θ ∈Ω

1(M) such that dθ is symplectic, and the vector field Z characterized by the property thatιZ dθ = θ points outward along ∂M . This implies that α := θ |∂M is a contact form. If for t ≥ 0we denote by ζt : M → M the time-t flow of the vector field −Z , then for some ε0 > 0 the mapΨ : (1− ε0, 1] × ∂M ,→ M defined by Ψ(r, m) = ζ− ln r(m) is an embedding with the propertythat Ψ∗θ = rα. Then the Liouville completion (M , θ ) may be defined as

M =M ∪(1− ε0,∞)× ∂M

m∼ Ψ−1(m) for m ∈ Im(Ψ)θ |M = θ , θ |(1−ε0,∞)×∂M = rα.

In particular (M , dθ ) is a symplectic manifold.Hereinafter we will identify (1 − ε0, 1] × ∂M with a subset of M ; it should be understood

that we are using Ψ to make this identification.

Definition 2.1. A filling Lagrangian in (M ,θ ) is a compact dim M

2-dimensional submanifold L of

M with boundary ∂ L, such that:

(i) ∂ L ⊂ ∂M .(ii) For some εL with 0< εL < ε0 we have (with respect to Ψ)

L ∩[1− εL , 1]× ∂M

= [1− εL , 1]× ∂ L

(iii) For some smooth function h: L→ Rwhich vanishes to infinite order along ∂ L, we haveθ |L = dh.

Of course, (iii) implies that L is an exact Lagrangian submanifold, and (ii) and (iii) togetherimply that in fact θ |L∩([1−εL ,1]×∂ M) = 0. Any filling Lagrangian L ⊂ M gives rise to a properly

embedded, exact Lagrangian submanifold without boundary L ⊂ M , namely

L = L ∪ ([1,∞)× ∂M).

We will refer to L as the completion of L.

8 MICHAEL USHER

The standard example of a filling Lagrangian occurs when M = D∗N is the disk cotangentbundle of some smooth manifold N and θ is the tautological one-form; then we may take L equalto the disk conormal bundle of any compact submanifold of N . (Indeed in this case θ |L = 0.)The name “filling Lagrangian” refers to the fact that L provides what is sometimes called a fillingfor the Legendrian submanifold ∂ L ⊂ ∂M .

The results of this paper make use of a Floer complexC F

c( L0, L1; H),∂J

that may be asso-ciated to data of the following sort:

(I) Completions L0, L1 ⊂ M of a pair of filling Lagrangians L0, L1 ⊂ M such that ∂ L0∩∂ L1 =

∅.(II) A smooth, compactly supported function H : [0, 1] × M → R, such that the time-one

map φ1H

of the Hamiltonian flow of H has the property that φ1H( L0) is transverse to L1.

(III) An element c ∈ π0(P ( L0, L1)), whereP ( L0, L1) is the space of piecewise-smooth pathsγ: [0, 1]→ M such that γ(0) ∈ L0,γ(1) ∈ L1.

(IV) A suitably generic path J= Jtt∈[0,1] of dθ -compatible almost complex structures on M

such that, for some real number RJ > 1−ε0, we have Jt = J0 for all t on [RJ,∞)×∂M ,and α J0 = dr on [RJ,∞)× ∂M .

Given L0, L1, and c ∈ π0(P ( L0, L1)), fix (independently of H and J) a path γc: [0, 1] →

M which represents the class c, and fix a symplectic trivialization τc: γ∗

cT M → [0, 1] × R2n

which sends Tγ(0) L0 to 0 × Rn × ~0 and Tγ(1) L1 to 1 × Rn × ~0. Formally speaking,C F

c(L0, L1; H),∂J

is the Morse complex of the function AH : c→ R (where again c is a pathcomponent of P ( L0, L1)), defined by

AH(γ) = −∫

[0,1]2u∗dθ +

∫ 1

0

H(t,γ(t))d t

where u: [0, 1]2 → M is a smooth map with u(0, t) = γc(t), u(1, t) = γ(t), u(s, 0) ∈ L0, and

u(s, 1) ∈ L1 for all s, t ∈ [0, 1] (the fact that the Li are exact makes AH independent of thechoice of such u). A critical point γ ofAH has γ′(t) = XH(t,γ(t)) for all t; thus critical points ofAH are in natural bijection with intersection points γ(0) ∈ L0 ∩ (φ1

H)−1( L1). Since φ1

His equal

to the identity outside of a compact set and since L0 ∩ L1 ∩ ([1,∞)× ∂M) = ∅, the assumedtransversality of φ1

H( L0) and L1 implies that there are only finitely many such points γ(0), and

so only finitely many elements γ in the critical locus C ri t(AH) ofAH .There is a homomorphism µ

c: π1(c,γc)→ Zwhich assigns to the homotopy class of a smooth

map u: (S1× [0, 1], S1×0, S1×1)→ (M , L0, L1) the difference of the Maslov indices of theloops of Lagrangian subspaces (u|S1×1)

∗T L1 and (u|S1×0)∗T L0 with respect to an arbitrary

symplectic trivialization of u∗T M . Let Nc∈ Z be the positive generator of the image of µ

c(if

µc= 0—as will be the case in our main application—we set N

c= 0).

Given these data, to each γ ∈ C ri t(AH) representing the class c we may associate a Maslov-type index µ(γ) ∈ Z/N

cZ. Namely, choose an arbitrary piecewise C1 map v : [0, 1]2 → M

such that v(0, t) = γc(t), v(1, t) = γ(t), v(s, 0) ∈ L0, and v(s, 1) ∈ L1 for all s, t ∈ [0, 1], and

choose a symplectic trivialization of v∗T M which restricts over 0 × [0, 1] to the trivializationτc

and which identifies each Tv(s,i) Li with Rn × ~0. With respect to this trivialization, the pathΓ (t) = (φ t

H)∗Tγ(0) L0 is a path in the Lagrangian Grassmannian such that Γ (0) = Rn × ~0 and

Γ (1) is transverse to Rn × ~0. We let

(6) µ(γ) =n

2−µRS(Γ ,Rn × ~0) mod N

c


where µRS(Γ ,Rn × ~0) is the Robbin–Salamon–Maslov index of the path Γ with respect toRn×~0 as defined in [RS93, Section 2]. Since µRS(Γ ,Rn×~0) receives an initial contributionof one-half of the signature of the crossing form of Γ with Rn × ~0 at t = 0, and this signatureis congruent to n modulo 2, n

2−µRS(Γ ,Rn×~0) is indeed an integer; the subsequent reduction

modulo Nc

removes the dependence of µ(γ) on the homotopy v from γc

to γ, so that µ(γ) ∈ ZNcZ

indeed depends only on γ (given the choices of γc

and τc

that were made at the outset). Ournormalization is designed so that if L0 = L1 is the zero section of the cotangent bundle T ∗N andH is the pullback of a C2-small Morse function on N , so that elements of C ri t(AH) are constantpaths γp at critical points p of H, then the Maslov index µ(γp) coincides with the Morse indexof p.

Now for k ∈ Z/NcZ let C F

c,k( L0, L1; H) be the Z/2Z-vector space generated by those γ ∈C ri t(AH) representing the homotopy class c with µ(γ) = k. For λ ∈ R let C Fλ

c,k( L0, L1; H) ≤C F

c,k( L0, L1; H) be the subspace generated by those γ which additionally haveAH(γ)≤ λ.Given a path of almost complex structures J = Jtt∈[0,1] as in (IV) above we consider the

associated negative gradient flow equation forAH , for a map u: R× [0, 1]→ M :

(7)∂ u

∂ s+ Jt(u(s, t))

∂ u

∂ t− XH(t, u(s, t))

= 0 u(s, 0) ∈ L0, u(s, 1) ∈ L1 for all s ∈ R.

For any finite-energy solution to (7) there are γ± ∈ C ri t(AH) such that u(s, ·)→ γ± uniformlyin t as s → ±∞. Note that our assumptions on H and on the Li ensure that γ− and γ+ bothhave image contained in M \([RH ,∞)×∂M), where RH ≥ RJ is chosen so large that H vanishesidentically on [0, 1]× [RH ,∞)× ∂M . In fact, our assumptions on J together with a maximumprinciple ensure that any finite-energy solution u to (7) must have image contained entirely inM \ ((RH ,∞)×∂M), as can be seen for instance from [Oh01, Theorem 2.1] or [AS10, Sections7c,7d].

In view of this maximum principle and of the fact that bubbling is prevented by the exactnessof the symplectic form on M and of the Lagrangian submanifolds L0, L1, the standard construc-tion of the Floer boundary operator as in [F89], [Oh97] gives for suitably generic J a map∂J : C F

c( L0, L1; H)→ C F

c( L0, L1; H) which counts finite-energy index-one solutions to (7), and

satisfies ∂J ∂J = 0. Moreover for each λ ∈ R and k ∈ Z/NcZ we have

∂J(C Fλc,k( L0, L1; H)) ⊂ C Fλ

c,k−1( L0, L1; H).

Thus (C Fc( L0, L1; H),∂J) is a Z/N

cZ-graded, R-filtered chain complex of Z/2Z-vector spaces.

For any element c =∑

i aiγi ∈ C Fc( L0, L1; H) (where ai ∈ Z/2Z and γi ∈ C ri t(AH)), write

ℓ(c) =maxAH(γi)|ai 6= 0,where the maximum of the empty set is defined to be −∞. Thus ℓ(∂Jc) ≤ ℓ(c) for all c ∈C F

c( L0, L1; H).

Choose a monotone increasing smooth function β : R→ [0, 1] such that β(s) = 0 for s ≤ −1and β(s) = 1 for s ≥ 1. Given two pairs (H−,J− = J−,t), (H+,J+ = J+,t) as in (II), (IV),define H : R× [0, 1]× M → R by H(s, t, m) = β(s)H−(t, m) + (1− β(s))H+(m). For a suitablygeneric family of almost complex structures Js,t as in (IV) which interpolate between J−,t andJ+,t , counting isolated solutions u: R× [0, 1]→ M to

∂ u

∂ s+ Js,t(u(s, t))

∂ u

∂ t− XH(s,·)(t, u(s, t))

= 0, u(R× 0) ⊂ L0, u(R× 1) ⊂ L1

10 MICHAEL USHER

gives rise to a chain map

ΦH−H+: C F

c( L0, L1; H−)→ C F

c( L0, L1; H+).

A standard estimate (see, e.g., [Oh97, p. 564], noting the different sign conventions) gives

ℓ(ΦH−H+c)≤ ℓ(c) + ‖H+ − H−‖

for all c ∈ C Fc( L0, L1; H−), where ‖ · ‖ denotes the Hofer norm

‖H‖=∫ 1

0

max

MH(t, ·)−min

MH(t, ·)

d t.

Moreover by considering appropriate homotopies of homotopies (see e.g., [U11a, Proposition2.2] and the text preceding it for details in the essentially identical context of Hamiltonian Floertheory), one obtains maps K± : C F

c( L0, L1; H±)→ C F

c( L0, L1; H±) such that

ΦH+H− ΦH−H+− 1= ∂J− K− +K− ∂J− , ΦH−H+

ΦH+H− − 1= ∂J+ K+ +K+ ∂J+and, for all c ∈ C F

c( L0, L1; H±),

ℓ(K±c)≤ ℓ(c) + ‖H+ − H−‖.In other words, in the language of [U11b, Definition 3.7], the Z/N

cZ-graded, R-filtered com-

plexes (C Fc( L0, L1; H−),∂J−) and (C F

c( L0, L1; H+),∂J+) are “‖H+ − H−‖-quasiequivalent.”

We define the boundary depth

βc,k( L0, L1; H) = inf

nβ ≥ 0(∀λ ∈ R)(Im∂J)∩ C Fλ

c,k( L0, L1; H) ⊂ ∂JC F

λ+β

c,k+1( L0, L1; H)o

.

[U11b, Proposition 3.8] shows that we have

(8) |βc,k( L0, L1; H+)− βc,k( L0, L1; H−)| ≤ ‖H+ − H−‖.

(In particular βc,k( L0, L1; H) is independent of the generic family of almost complex structures

as in (IV) used to define it.) A priori, βc,k( L0, L1; H) has only been defined when H obeys

(II), but (8) allows us to extend this definition continuously to arbitrary smooth (or even justcontinuous) compactly supported functions H : [0, 1]× M → R.

Now generators for the Floer complex C Fc( L0, L1; H) correspond to certain intersection points

of L0 with (φ1H)−1( L1), while the definition of the filtration on the complex and hence of the

boundary depth βc,k( L0, L1; H) at least appear to depend on the particular Hamiltonian function

H generating the time-one map φ1H

. In fact we will presently see that, much like the situationin [U11b, Section 6], modulo shifts in the homotopy and grading data c and k the boundarydepth actually only depends on the Lagrangian submanifolds L0 and (φ1

H)−1( L1).

To set this up, consider any Hamiltonian isotopy ψt : M → M (0≤ t ≤ 1) with ψ0 = 1M andsuch that ψ1( L1) = L1. For a path γ ∈ P ( L0, L1), define a new path Ψγ: [0, 1]→ M by

(Ψγ)(t) =ψt(γ(t)).

The fact that ψ1( L1) = L1 implies that Ψγ ∈ P ( L0, L1). This induces a map

Ψ∗ : π0(P ( L0, L1))→ π0(P ( L0, L1))

defined byΨ∗[γ] = [Ψγ]. Where HamL1is the subgroup of Ham(M , dθ ) consisting of Hamilton-

ian diffeomorphisms which preserve L1, the map Ψ∗ evidently depends only on the relative ho-motopy class [Ψ] ∈ π1(Ham(M , dθ ), Ham L1

, 1M ). Thus we have an action ofπ1(Ham(M , dθ ), HamL1, 1M )

on π0(P ( L0, L1)).


Remark 2.2. In our main application it will hold that, for any compact set K ⊂ M , every class inπ0(P ( L0, L1)) is represented by a path which is disjoint from K . Since our Hamiltonian isotopiesare compactly supported it follows that in this case the action of π1(Ham(M , dθ ), HamL1

, 1M )

on π0(P ( L0, L1)) is trivial.

If two Hamiltonians G, H : [0, 1]× M → R have the property that (φ1H)−1( L1) = (φ

1G)−1( L1),

define ψt = φtH (φ t

G)−1, so that ψ1( L1) = L1. If F : [0, 1] × M → R is the Hamiltonian

generating the isotopy ψt, we will have

(9) G(t, m) = (H − F)(t,ψt(m))

for (t, m) ∈ [0, 1]× M .

Proposition 2.3. If G, H, F,ψt are as above and if φ1H( L0) is transverse to L1, then for each

c ∈ π0(P ( L0, L1)) there are µc∈ Z/N

cZ, λ

c∈ R, and paths of almost complex strucures J, J′ as

in (IV) such that there is an isomorphism of chain complexes

Φ : (C Fc( L0, L1; G),∂J)→ (C FΨ∗c( L0, L1; H),∂J′)

with the property that, for each k ∈ Z/NcZ and λ ∈ R,

Φ

C Fλ

c,k( L0, L1; G)= C F

λ+λc

Ψ∗c,k+µc

( L0, L1; H).

Proof. The proof is essentially the same as that of [U11b, Proposition 6.2]. First note that apath γ: [0, 1]→ M represents the class c ∈ π0(P ( L0, L1)) and belongs to C ri t(AG) if and onlyif the curve Ψγ represents the class Ψ∗c and belongs to C ri t(AH). So we can define the map Φon generators by setting Φ(γ) = Ψγ; obviously Φ is an isomorphism of vector spaces.

For general maps u: R× [0, 1]→ M and K : [0, 1]× M → R and families of almost complexstructures J= Jt0≤t≤1 we denote

∂J,Ku=∂ u

∂ s+ Jt

∂ u

∂ t− XK

;

this is a section of u∗T M . If, given J as in (IV), we define J′ = J ′t0≤t≤1 by J ′

t= (ψt)∗Jt(ψ

−1t)∗,

a routine computation shows that, for all (s, t) ∈ R× S1,

(ψt)∗(∂J,Gu)(s,t)=∂J′,H(Ψu)(s,t)

where (Ψu)(s, t) =ψt(u(s, t)). Of course we have u(R×i) ⊂ Li if and only if (Ψu)(R×i) ⊂Li , since ψ0 is the identity and ψ1( L1) = L1. So the map u 7→ Ψu sends the moduli spacesof flowlines used to define the differential ∂J on C F

c( L0, L1; G) bijectively to the corresponding

(via Φ) moduli spaces used to define ∂J′ on C Fc( L0, L1; H) (and moreover preserves Fredholm

regularity of these solutions). This proves thatΦ is an isomorphism of chain complexes, providedof course that J has been chosen generically so as to ensure that ∂J is well-defined and satisfiesthe usual properties.

It remains to prove the statement about the effect of Φ on gradings and filtrations. Recall that,to define the action functionals AG ,AH , we have chosen representatives γ

cfor each class c ∈

π0(P ( L0, L1)). For each c ∈ π0(P ( L0, L1), choose once and for all a map w : [0, 1]×[0, 1]→ M

such that w([0, 1]× i) ⊂ Li for i = 0, 1, while w(0, ·) = γΨ∗c and w(1, ·) = Ψγc.

If γ is a representative of c, let v : [0, 1] × [0, 1] → M be such that v([0, 1] × i) ⊂ Li fori = 0, 1, v(0, ·) = γ

c, and v(1, ·) = γ. Also define (Ψv)(s, t) = ψt(v(s, t)). Then concatenating

12 MICHAEL USHER

w and Ψv gives a homotopy from γΨ∗c to Ψγ. We therefore have, using (9) and the fact that theψt are symplectomorphisms

AH(Ψγ) = −∫

[0,1]2w∗dθ −∫

[0,1]2(Ψv)∗dθ +

∫ 1

0

H(t,ψt(γ(t)))d t

= −∫

[0,1]2w∗dθ −∫

[0,1]2v∗dθ −∫ 1

0

∫ 1

0

(dF)ψt (v(s,t))

∂ (Ψv)

∂ s

dsd t +

∫ 1

0

H(t,ψt(γ(t)))d t

= −∫

[0,1]2w∗dθ −∫

[0,1]2v∗θ −∫ 1

0

(F(t,ψt(γ(t)))− F(t,Ψγc))d t +

∫ 1

0

H(t,ψt(γ(t)))d t

=AF (γΨ∗c)−∫

[0,1]2v∗dθ +

∫ 1

0

(H − F)(t,ψt(γ(t)))d t

=AF (γΨ∗c) +AG(γ).

This proves that our chain isomorphismΦ obeys ℓ(Φc) = ℓ(c)+AF (γΨ∗c) for all c ∈ C Fc( L0, L1; H),

so we may set λc=AF (γΨ∗c) in the statement of the proposition.

As for the gradings, by using a trivialization over the concatenation of the above maps w andΨv to compute µ(Ψγ), one may verify that µ(Ψγ)−µ(γ) is given as follows. Let τw : w∗T M →[0, 1]2×R2n be a symplectic trivialization which restricts to our fixed trivialization τΨ∗c over γΨ∗cand which sends (w∗T Li)(s,i) toRn×~0 for all s ∈ [0, 1] and i = 0, 1. We then have a Lagrangiansubbundle Lw = (τ

w)−1(1 × [0, 1] × Rn × ~0) of (Ψγc)∗T M , which coincides over i = 0, 1

with T(Ψγ)(i) Li . Meanwhile, we have a trivialization Ψ∗τc: (Ψγ

c)∗T M → [0, 1]×R2n obtained by

pushing forward the fixed trivialization τc

of γ∗cT M in the obvious way, and Ψ∗τc

(T(Ψγ)(i) Li) =

Rn×~0. Ψ∗τc(Lw) thus defines a loop of Lagrangian subspaces ofR2n, and using the catenation

property of the Robbin–Salamon–Maslov index µ(Ψγ) − µ(γ) can be seen to agree with theMaslov index of this loop. So we may define µ

cas the mod N

creduction of the Maslov index of

Ψ∗τc(Lw), completing the proof.

Recall from the introduction that for any closed subset S ⊂ M we define

L (S) = φ(S)|φ ∈ Ham(M , dθ ),and for S1, S2 ∈ L (S)

δ(S1, S2) = inf

(∫ 1

0

max

MH(t, ·)−min

MH(t, ·)

d t

φ1H(S1) = S2, H ∈ C∞

c([0, 1]× M)

).

where C∞c([0, 1]× M) denotes the space of compactly-supported smooth real-valued functions

on [0, 1]× M .If L0, L1 are two filling Lagrangians in M , and if Λ ∈ L ( L1), define

(10) B( L0,Λ) = supc,kβc,k( L0, L1; H) for any H ∈ C∞

c([0, 1]× M) with (φ1

H)−1( L1) = Λ.

Corollary 2.4. B( L0,Λ) is independent of the choice of H used to define it. Moreover, for any

Λ,Λ′ ∈ L ( L1),

δ(Λ,Λ′)≥ |B( L0,Λ)− B( L0,Λ′)|.


Proof. Suppose for the moment that L0 is transverse toΛ, so that if we choose any H ∈ C∞c([0, 1]×

M) such that (φ1H)−1( L1) = Λ then we have a well-defined Floer complex (C F( L0, L1; H),∂J),

with the boundary depth βc,k( L0, L1; H) independent of the choice of J. If G ∈ C∞

c([0, 1]× M)

also has (φ1G)−1( L1) = Λ, it follows directly from Proposition 2.3 that for each c ∈ π0(P ( L0, L1))

there is µc∈ Z/N

cZ such that, for all k ∈ Z/N

cZ, β

c,k( L0, L1; G) = βΨ∗c,k+µc

( L0, L1; H) (whereΨ∗ : π0(P ( L0, L1))→ π0(P ( L0, L1)) is a bijection). In particular the suprema of β over all (c, k)

are identical for G and H, confirming that B( L0,Λ) is independent of the choice of H used todefine it, at least if L0 and Λ are transverse.

Now suppose that Λ,Λ′ ∈ L ( L1) are both transverse to L0, with Λ = (φ1H)−1( L1) and Λ′ =

φ1F(Λ). Then where G(t, m) = H(t, m)− F(t,φ t

F((φ t

H)−1(m))) we have φ t

G= φ t

H (φ t

F)−1 and

so (φ1G)−1( L1) = Λ

′. Note that F(t, m) = (H − G)(t,φ tG(m)). By (8) we have, for all c, k,

|βc,k( L0, L1; H)− β

c,k( L0, L1; G)| ≤∫ 1

0

max

M(H − G)(t, ·)−min

M(H − G)(t, ·)

d t

=

∫ 1

0

max

MF(t, ·)−min

MF(t, ·)

d t

Since this holds for all c, k it follows that |B( L0,Λ)−B( L0,Λ′)| ≤∫ 1

0

maxM F(t, ·)−minM F(t, ·)

d t

But F ∈ C∞c([0, 1]× M)was arbitrary subject to the requirement that Λ′ = φ1

F(Λ), so this proves

that |B( L0,Λ)− B( L0,Λ′)| ≤ δ(Λ,Λ′).Finally, the case in which Λ and/or Λ′ is not transverse to L0 follows straightforwardly by

a limiting argument from the transverse case, by using C0-small Hamiltonians F, F ′ such thatφ1

F(Λ),φ1

F ′(Λ′) are transverse to L0 (and bearing in mind that, in the nontransverse case, β

c,kwas defined as a limit using (8)).

3. REPARAMETRIZED GEODESIC FLOWS

To begin the analysis leading to our main results, we now specialize the discussion of Section2 to the following situation. Let (N , g) be a compact Riemannian manifold, let x1 ∈ N , and letQ ⊂ N be a compact submanifold not containing x1. The conormal bundle of Q is

ν∗Q =(x , p) ∈ T ∗N |Qp|TxQ = 0

Under the identification of T ∗N with T N given by g, we have an exponential map exp: ν∗Q→N ; recall that x1 is said to be a focal point of Q if x1 is a critical value of this map. In particular,by Sard’s theorem, almost every x1 ∈ N is not a focal point of Q. In the case that Q is a singletonx0 (so ν∗Q = T ∗

x0N), a focal point of x0 is also known as a conjugate point of x0.

The g-identification of T ∗N and T N gives an inner product ⟨·, ·⟩ and norm | · | on the fibersof T ∗N . Our Liouville domain will be the disk bundle M = D∗N = (x , p) ∈ T ∗N ||p| ≤ 1,endowed with the canonical one-form θ(x ,p)(v) = p(π∗v), with boundary S∗N = (x , p)||p|= 1.The completion M is then just the cotangent bundle T ∗N with its canonical one-form θ . Morespecifically, where 0N is the zero section of T ∗N , the Liouville completion process identifiesT ∗N \ 0N with (0,∞)r × S∗N , with the canonical one-form θ identified with rα where

r(x , p) = |p| α(x ,p)(v) =p

|p| (π∗v)

(i.e. α is the pullback of the contact form θ |S∗N by the standard projection T ∗N \ 0N → S∗N).

14 MICHAEL USHER

Our filling Lagrangians L0, L1 ⊂ M = D∗N will be given by the disk conormal bundle L0 =

ν∗Q ∩ D∗N and the disk cotangent fiber L1 = T ∗x1

N ∩ D∗N , where x1 ∈ N and Q ⊂ N \ x1 isa compact submanifold such that x1 is not a focal point of Q. So in particular θ |L0

= θ |L1= 0

and, in the notation of Section 2, L0 = ν∗Q and L1 = T ∗

x1N . We will consider Hamiltonians

H : [0, 1]× T ∗N → R of the following special form:

H f (t, (x , p)) = f (|p|) where f ∈ C∞([0,∞)), supp( f ) ⊂ [0, R), supp( f ′) ⊂ (0, R)

for some R> 0.So where ψt : S∗N → S∗N is the Reeb flow of α = θ |S∗N on S∗N , and where φ t

H f: T ∗N →

T ∗N is the Hamiltonian flow of H f , we see that φ tH f

restricts to 0N as the identity (since f ′ is

supported away from 0), while on T ∗N \ 0N∼= (0,∞)× S∗N , we have

(11) φ tH f(r, z) = (r,ψ f ′(r)t(z))

Now the Reeb flow ψt : S∗N → S∗N is well-known and easily-verified to be given by thegeodesic flow: using the identification of S∗N with the sphere tangent bundle given by themetric, for z = (x , p) ∈ S∗N (so |p|= 1) ψt((x , p)) is given by the position and velocity at timet of the geodesic whose initial position and velocity are x and p.

In particular, the restriction of φ1H f

to L0 = ν∗Q sends (x , p) ∈ T ∗N where p|TxQ = 0 to the

pair (γ( f ′(|p|)), |p|γ′( f ′(|p|))), where γ is the geodesic with initial position and velocity givenby x0 and p

|p| . In order to set up the Floer complex C F(ν∗Q, T ∗x1

N ; H), we require that φ1H f(ν∗Q)

be transverse to T ∗x1

N . This holds if and only if, where π: T ∗N → N is the bundle projection,

π φ1H f

: ν∗Q→ N has x1 as a regular value. Now we have

π φ1H f(p) = exp

f ′(|p|)p

|p|

Since x1 is assumed to not be a focal point of Q, x1 is a regular value of exp: ν∗Q → N ;consequently x1 will be a regular value of π φ1

H f: ν∗Q→ N provided that

(12)If r ∈ R and if there is a geodesic γ such that γ′(0) ∈ ν∗Q, γ(1) = x1,

and γ has length ℓ= | f ′(r)|, then f ′′(r) 6= 0

(here we use the g-identification of T ∗N with T N to view ν∗Q as a subset of T N).Where we denote the sphere conormal bundle by Sν∗Q = S∗N∩ν∗Q and the sphere cotangent

fiber by S∗x1

N = S∗N ∩ T ∗x1

N , let

B(Q, x1) = (τ, y) ∈ R× Sν∗Q|ψτ(y) ∈ S∗x1(N).

Thus (τ, (x , p)) ∈ B(Q, x1) if and only if the geodesic γ: R→ N with γ(0) = x and γ′(0) = p

has γ(τ) = x1; note that negative values of τ are permitted. So elements (τ, y) of B(Q, x1)

are in one-to-one correspondence with Reeb flowlines (for τ > 0) or negative Reeb flowlines(for τ < 0) of duration |τ| which begin on Sν∗Q and end on S∗

x1N . The action of the positive or

negative Reeb flowline η(τ,y) corresponding to (τ, y), defined as∫η(τ,y)

α, is in either case just

given by τ ∈ R.Nowπ0(P (ν∗Q, T ∗

x1N)) is in obvious bijection (via the bundle projection) with the setPN (Q, x1)

of homotopy classes of paths in N from Q to x1; this identification will be implicit in what fol-lows. For c ∈ π0(P (ν∗Q, T ∗

x1N)) let

Bc(Q, x1) = (τ, y) ∈B(Q, x1)|[π η(τ,y)] = c


where π: S∗N → N is the bundle projection and as in the last paragraph η(τ,y) is the positiveor negative Reeb flowline from Sν∗Q to S∗

x1N that corresponds to the pair (τ, y). Also let

B+c(Q, x1) = (τ, y) ∈B

c(x0, x1)|τ > 0

The cotangent bundle T ∗M has an involution h: T ∗M → T ∗M defined by h(x , p) = (x ,−p);this obeys h∗θ = −θ and restricts to an involution h: S∗M → S∗M obeying h∗α = −α. Thusthe Reeb flow ψtt∈R obeys hψt =ψ−t h. Moreover obviously h( Li) = Li and π h= π, inview of which (τ, y) ∈ B

c(Q, x1) if and only if (−τ, h(y)) ∈ B

c(Q, x1). Consequently, defining

h∗ : Bc(Q, x1)→Bc

(Q, x1) by h∗(τ, y) = (−τ, h(y)), we have

Bc(Q, x1) =B+c (Q, x1)

∐h∗B+

c(Q, x1)

We now set up the filtered Floer complex C Fc( L0, L1; H). We must first choose a basepoint

γc: [0, 1] → T ∗N for our homotopy class c ∈ π0(P ( L0, L1)); we take γ

cto be an arbitrary

path in the class c that is contained in the zero section 0N (and so necessarily has γc(0) =

(x0, 0),γc(1) = (x1, 0) for some x0 ∈ Q). For later use it will be convenient to assume also that

γc|[0,1/2] is constant.

With this choice of γc, since the 1-form θ vanishes on each of 0N , L0, L1, we see from Stokes’

theorem that, for any Hamiltonian H : [0, 1]× T ∗N → R, the action functional AH : c→ R isgiven by

AH(γ) = −∫

[0,1]

γ∗θ +

∫ 1

0

H(t,γ(t))d t

Specializing to the case of our Hamiltonians H f (t, p) = f (|p|), the critical points ofAH f: c→

R may be described as follows. Let

Cc( f ,Q, x1) = (r, y) ∈ (0,∞)× S∗

x0N |( f ′(r), y) ∈B

c(Q, x1)

The critical points γ: [0, 1]→ T ∗N for AH fwhich represent the homotopy class c are all con-

tained in T ∗N \ 0N , and with respect to our identification of T ∗N \ 0N with (0,∞) × S∗N areprecisely the curves of the form

γ(r,y)(t) = (r,ψ f ′(r)t(y))(r, y) ∈ C

c( f ,Q, x1)

Since θ = rα, we immediately see that

(13) AH f(γ(r,y)) = f (r)− r f ′(r)

3.1. Grading. We now discuss the grading on the Floer complex. The discussion is somewhatsimpler when the submanifold Q is a singleton x0, so we consider that case first, discussing thenecessary modifications for more general Q later. Thus from now through Proposition 3.1 weassume that L0 = T ∗

x0N , and as always L1 = T ∗

x1N , where the points x0 and x1 are nonconjugate.

First we must also choose a symplectic trivialization for γ∗cT (T ∗N) where γ

c: [0, 1]→ 0N ⊂

T ∗N is the basepoint of c chosen earlier; we choose any trivialization with the property thateach of the cotangent fibers T ∗

γc(t)

N is mapped to Rn×0. Note that if v : [0, 1]×S1→ T ∗N is

any map such that v(i × S1) ⊂ Li for i = 0, 1, then we can trivialize v∗T M in such a way thatthe tangent spaces at v(s, t) to the cotangent fibers T ∗

π(v(s,t))N are all sent to Rn×~0. Thereforethe Maslov homomorphism µ

c: π1(c,γc)→ Z vanishes and so our grading will be by Z.

Before addressing the Maslov indices of the γ(r,y) we discuss the positive Reeb flowlinesη(τ,y) : [0,τ] → S∗N (defined by η(τ,y)(s) = ψs(y)) for (τ, y) ∈ B+

c(x0, x1). The contact

16 MICHAEL USHER

distribution ξ= kerα on S∗N is a rank-(2n−2) symplectic vector bundle over S∗N (with fiber-wise symplectic form given by dα), and for all s the tangent space at η(τ,y)(s) to the spherecotangent fiber S∗

π(η(τ,y)(s))N is a Lagrangian subspace of ξ. So we can symplectically trivialize

η∗(τ,y)ξ in a manner which sends these vertical Lagrangian subspaces to Rn−1×~0. With respect

to this trivialization, the path s 7→ (ψs)∗TzS∗x0

N defines a path of Lagrangian subspaces of R2n−2.

This path has a Robbin-Salamon-Maslov index (with respect to Rn−1 ×~0) [RS93]; we denotethis index by ν(η(τ,y)).

The index ν(η(τ,y)) is a sum of contributions corresponding to the intersections of (ψs)∗TyS∗x0

N

with Tψs(y)S∗π(ψs(y))

N as s varies from 0 to τ. Recalling that ψs is the time-s geodesic flow onS∗N , the restriction of the linearization (ψs)∗ to TyS∗

x0N is given by the derivative of the time-s

version of the exponential map. So (ψs)∗ maps v ∈ TyS∗x0

N to the pair (Jv(s), J ′v(s)) where Jv

is the Jacobi field along the geodesic π η(τ,y) having Jv(0) = 0 and J ′v(0) = v. Consequently

the intersections (ψs)∗TyS∗x0

N ∩ Tψs(y)S∗π(ψs(y))

N may be identified with the space of normalJacobi fields along π η(τ,y)|[0,s] which vanish at times 0 and s. The crossing form of [RS93,Theorem 1.1] at time s is then given by Q((0, J ′

v(s))) = dα((0, J ′

v(s)), (J ′

v(s), J ′′

v(s))) = |J ′

v(s)|2;

in particular the crossing form is positive definite.So according to the definition in [RS93], the Maslov index is given by

ν(η(τ,y)) =n− 1

2+∑

0<s<τ

dim(ψs)∗TyS∗

x0N ∩ Tψs(y)

S∗π(ψs(y))N

(the n−12

is the contribution from s = 0; there is no contribution from s = τ since x0 and x1

are nonconjugate). In other words ν(η(τ,y))− n−12

is the number of conjugate points along thegeodesic π η(τ,y), counted with multiplicity. According to the Morse Index Theorem [Mo34,Theorem V.15.2], this latter quantity is precisely the Morse index of the geodesic πη(τ,y) (afterit is reparametrized to have duration 1), considered as a critical point of the energy functionalη 7→∫|η′|2 on paths from x0 to x1; thus

(14) For (τ, y) ∈B+c(x0, x1), ν(η(τ,y)) =

n− 1

2+Morse(π η(τ,y))

where “Morse” denotes the aforementioned Morse index.Now consider a critical point γ(r0,y) ofAH f

with f ′(r0)> 0. Thus, identifying T ∗N \ 0N with(0,∞)× S∗N in our usual way,

γ(r0,y)(s) = (r0,η( f ′(r0),y)( f′(r0)s)) = (r0,ψ f ′(r0)s

(y))

To determine the grading of γ(r0,y) we must first determine the Maslov index of the path s 7→(φs

H f)∗T(r0,y)T

∗x0

N relative to the path of cotangent fibers s 7→ Tγ(r0,y)(s)T ∗π(γ(r0,y))

N .

The splitting T ∗N \ 0N∼= (0,∞)× S∗N gives rise to a splitting

T(r0,y)T∗x0

N ∼= R∂r ⊕ TyS∗x0

N

If z ∈ TyS∗x0

N then (φsH f)∗(0, z) = (0, (ψ f ′(r0)s

)∗z). Meanwhile the element ∂r ∈ T(r0,y)T∗x0

N has

(φsH f)∗∂r =∂r , s f ′′(r0)η

′( f ′(r0),y)

( f ′(r0)s)

.

In particular (under the transversality condition (12)) if s > 0 then π∗(φsH f)∗∂r 6= 0, and so

the intersections of (φsH f)∗T(r0,y)T

∗x0

N with Tγ(r0,y)(s)T ∗π(γ(r0,y))

N are in one-to-one correspondence


with the intersections of (ψ f ′(r0)s)∗TyS∗

x0N with Tψ f ′(r0)s(y)

S∗π(ψ f ′(r0)s(y))

N ; moreover the Robbin–

Salamon crossing forms are identical under this correspondence. On the other hand at s = 0the vector ∂r provides a single additional dimension of intersection for γ(r0,y) in comparison toη( f ′(r0),y). We have

d

ds

s=0

(φsH f)∗∂r = (0, f ′′(r0)η

′( f ′(r0),y)

(0))

with respect to the splitting (0,∞) × S∗N . Now with the respect to the (horizontal,vertical)splitting of the sphere (co)tangent bundle we have η′

( f ′(r0),y)(0) = (y, 0). Meanwhile in the

(horizontal,vertical) splitting the tangent vector ∂r at γr0,y(0) corresponds to (0, y). Thus thecrossing form evaluates on ∂r at s = 0 as Q(∂r) =ω((0, y), ( f ′′(r0)y, 0)) = f ′′(r0)|y |2.

Consequently when f ′(r0)> 0 the Maslov index of s 7→ (φsH f)∗T(r0,y)T

∗x0

N is ν(η( f ′(r0),y)) +12

if f ′′(r0)> 0 and is ν(η( f ′(r0),y))−12

if f ′′(r0)< 0.There remains the case that f ′(r0) < 0. In this case, where h: T ∗N → T ∗N is again given

(with respect to the (horizontal,vertical) splitting) by h(x , p) = (x ,−p), we have (with respectto the splitting T ∗N \ 0N = (0,∞)× S∗N),

γ(r0,y)(s) = (r0,ψ f ′(r0)s(y)) =r0, h(ψ− f ′(r0)s

(h(y)))= (r0, h(η(− f ′(r0),h(y))(− f ′(r0)s))).

So with the exception of the term ± 12

coming from f ′′(r0) at s = 0, the contributions tothe Maslov index of (φs

H f)∗T(r0,y)T

∗x0

N correspond to the contributions to the Maslov index of

s 7→ψ− f ′(r0)s(h(y)) (i.e., to ν(η(− f ′(r0),h(y)))); however the conjugation by the antisymplectic in-

volution h negates the crossing forms and so causes the respective contributions to be oppositeto each other. Meanwhile the same argument as in the previous case shows that the contributionof ∂r at s = 0 is 1

2if f ′′(r0)> 0 and − 1

2if f ′′(r0)< 0.

So when f ′(r0)< 0 the Maslov index of s 7→ (φsH f)∗T(r0,y)T

∗x0

N is−ν(η− f ′(r0),y)+12

if f ′′(r0)>

0 and is −ν(η− f ′(r0),y)−12

if f ′′(r0)< 0.In any case, projecting down to N we see that πγ(r0,y) and πη(| f ′(r0)|,y) represent the same

geodesic (modulo positive constant time rescaling) from x0 to x1. So we have, in view of (14)and our grading conventions (6):

Proposition 3.1. For each element (r, y) ∈ Cc( f , x0, x1), the corresponding element γ(r,y) ∈

C ri t(AH f) has Floer-theoretic grading given by

µ(γ(r,y)) =

−Morse(π γ(r,y)) f ′(r)> 0, f ′′(r)> 01−Morse(π γ(r,y)) f ′(r)> 0, f ′′(r)< 0n− 1+Morse(π γ(r,y)) f ′(r)< 0, f ′′(r)> 0n+Morse(π γ(r,y)) f ′(r)< 0, f ′′(r)< 0

We now turn to the more general situation in which the submanifold L0 = T ∗x0

N is replacedby the conormal bundle ν∗Q of a smooth, compact d-dimensional submanifold Q of N . Wecontinue to set L1 = T ∗

x1N , and now require that x1 is not a focal point of Q (in other words, is

not a critical value of the restriction of the exponential map to ν∗Q).A few subtleties arise in this extension because the conormal bundle ν∗Q typically has more

complicated geometry than a cotangent fiber. In particular the second fundamental form of Q

gives rise, for every (x , p) ∈ T ∗N |Q, to a symmetric “shape operator” Σ(x ,p) : TxQ→ TxQ definedby the property that, for v, w ∈ TxQ,

g(Σ(x ,p)v, w) = p(∇vW )⊥

18 MICHAEL USHER

where W is a vector field on a neighborhood of x in N which is tangent to Q and has W (x) = w,∇ is the Levi-Civita connection associated to g on N , and ⊥ denotes projection of a vector inT N |Q to the g-orthogonal complement TQ⊥ of TQ. We extend Σ(x ,p) to a symmetric operatorTx N → Tx N by setting it equal to zero on TxQ⊥.

The Levi-Civita connection ∇ (together with the g-identification of T ∗N with T N) gives us asplitting T (T ∗N) = T vt⊕T hor where T vt is the vertical tangent space of the projection T ∗N → N

and where, for a section s : N → T ∗N with s(x) = (x , p) and for v ∈ Tx N , we have s∗v ∈ T hor(x ,p)

if and only if, viewing s as a vector field, (∇vs)(x) = 0. For (x , p) ∈ T ∗N , we may identify bothT hor(x ,p) and T vt

(x ,p) with Tx N . Indeed, the projection-induced map π∗ : T(x ,p)T∗N → Tx N restricts

as an isomorphism T hor(x ,p)∼= Tx N ; we will denote the inverse of this isomorphism by v 7→ v# (so

v# is the horizontal lift of v in the standard sense). Meanwhile T vt(x ,p) is identified with T ∗

xN

by the identification of the tangent space to a vector space with the vector space, and T ∗xN is

identified with Tx N by the metric g. For v ∈ Tx N we will denote the corresponding element ofT vt(x ,p) by v.

Now for (x , p) ∈ ν∗Q, define

T hor(x ,p) = v# − (Σ(x ,p)v)

|v ∈ Tx N.

We then have a decomposition of bundles

T (T ∗N)|ν∗Q = T vt ⊕ T hor ,

and the fact that the Σ(x ,p) are symmetric translates to the statement that T hor is (like both T hor

and T vt) a Lagrangian subbundle of T (T ∗N)|ν∗Q with respect to the standard symplectic form

dθ . For v ∈ Tx N and (x , p) ∈ ν∗Q define ve# = v# − (Σ(x ,p)v)

; in other words, ve# is the unique

element of T hor(x ,p) that projects to v.

A straightforward calculation shows that the tangent space to the conormal bundle ν∗Q isgiven as follows:

(15) Tν∗Q = v|v ∈ TQ⊥ ⊕ ve#|v ∈ TQ =: (TQ⊥) ⊕ (TQ)e#.

Proposition 3.2. There is a smooth family of Lagrangian subbundles At ≤ T (T ∗N)|ν∗Q (0≤ t ≤ 1)

such that:

• A0 = Tν∗Q• A1 = T vt

• For each (x , p) ∈ ν∗Q, the Robbin–Salamon–Maslov index of the path (At)(x ,p)0≤t≤1with respect to the constant path T(x ,p)ν

∗Q is given by

µRS

(At)(x ,p), T(x ,p)ν

∗Q=

d

2

where d = dimQ.

Proof. Define an endomorphism J : T (T ∗N)|ν∗Q → T (T ∗N)|ν∗Q by J v = v# and J v# = −v.Thus J is a dθ -compatible almost complex structure on T (T ∗N)|ν∗Q, mapping T vt to T hor andvice versa.

For any s ∈ R, define an endomorphism esJ : T (T ∗N)|ν∗Q→ T (T ∗N)|ν∗Q by esJ w = (cos s)w+

(sin s)Jw. Then let

At = (TQ⊥) ⊕ eπt

2JTQe#


The At are easily seen to be Lagrangian, and we have A0 = Tν∗Q by (15), while A1 = (TQ⊥)⊕TQ = T vt . Meanwhile the product axiom for the Robbin–Salamon–Maslov index shows that

µRS((At)(x ,p), T(x ,p)ν∗Q) is equal to the Robbin–Salamon–Maslov index of the path t 7→ e

πt

2J

TQe#(x ,p)

with respect to TQe#(x ,p) in the 2d-dimensional symplectic vector space TQ

(x ,p) ⊕ TQe#(x ,p). The

only intersections relevant to this index are those at t = 0, and at t = 0 the crossing form iseasily seen to be positive definite, so that the index is indeed d

2.

With this preparation we can discuss the gradings of the Floer complexes C Fc(ν∗Q, T ∗

x1N ; H f ).

Recall that just before the start of Section 3.1 we chose a representative γc

of the homotopy classc ∈ π0(P (ν∗Q, T ∗

x1N)), such that γ

c(t) ∈ 0N for all t and γ

c|[0,1/2] is constant. For the grading we

must also fix a suitable trivialization τc

of γ∗cT (T ∗N). We choose this symplectic trivialization

τc: γ∗

cT (T ∗N)→ [0, 1]×R2n in such a way that

τ−1c(t ×Rn × ~0) =

¨(A2t)γ

c(0) 0≤ t ≤ 1

2T vtγc(t)

12≤ t ≤ 1

Any element ofπ1(c,γc) can be represented by a map v : S1×[0, 1]→ T ∗N such that v(θ , t) ∈ν∗Q for θ ∈ S1 and 0 ≤ t ≤ 1/2, while v(θ , 1) ∈ T ∗

x1N . Then v∗T (T ∗N) has a Lagrangian

subbundle Λ whose fiber over (θ , t) is given by (A2t)v(θ ,t) for 0 ≤ t ≤ 1/2 and by T vtv(θ ,t) for

1/2 ≤ t ≤ 1. Since Λ restricts to S1 × 0 as the pullback of Tν∗Q and to S1 × 1 as thepullback of T (T ∗

x1N), it follows that the Maslov homomorphism µ

c: π1(c,γc)→ Z vanishes on

[v]. Thus the grading of the Floer complexes C Fc(ν∗Q, T ∗

x1N ; H) is by Z.

Where Sν∗Q = S∗N ∩ν∗Q is the sphere conormal bundle of Q and where ψt : S∗N → S∗N isthe Reeb flow, recall the sets

Bc(Q, x1) = (τ, y) ∈ R× Sν∗Q|ψτ(y) ∈ S∗

x1N , [s 7→ψsτ(y)] ∈ c

andC

c( f ,Q, x1) = (r, y) ∈ (0,∞)× Sν∗Q|( f ′(r), y) ∈B

c(Q, x1).

The critical points ofAH f: c→ R are the paths

γ(r,y)(t) = rψ f ′(r)t(y) (r, y) ∈ Cc( f ,Q, x1)

It follows from our choice of τc

together with the catenation and homotopy properties of theRobbin–Salamon–Maslov index µRS for pairs of Lagrangian arcs ([RS93, Section 3]) that

(16) µ(γ(r,y)) =n

2−µRS(Tr yν

∗Q, t 7→ At) +µRS(t 7→ (φ tH f)∗Tr yν

∗Q, T vtφ t

H f(r y))

.

By Proposition 3.2 (and the fact that µRS is antisymmetric in its two arguments) we haveµRS(Tr yν

∗Q, t 7→ At) = − d

2. Meanwhile the computation of µRS(t 7→ (φ t

H f)∗Tr yν

∗Q, T vtφ t

H f(r y)) es-

sentially repeats what was done in the special case that Q = x0. Intersections of (φ tH f)∗Tr yν

∗Q

with T vtφ t

H f(r y)

where t > 0 correspond to focal points of Q along the geodesic π γ(r,y); these

contribute positively to µRS when f ′(r) > 0 and negatively to µRS when f ′(r) < 0, in eachcase according to their multiplicity. Now [Mo34, Theorem V.15.2] shows that the sum of themultiplicities of the focal points along π γ(r,y) is equal to the Morse index Morse(π γ(r,y))of the geodesic π γ(r,y), considered as a critical point of the energy functional on paths fromQ to x1. Thus µRS(t 7→ (φ t

H f)∗Tr yν

∗Q, T vtφ t

H f(r y)) is equal to si gn( f ′(r)) · Morse(π γ(r,y)) plus

one-half of the signature of the crossing form at t = 0. This latter signature is computed just

20 MICHAEL USHER

as in the previously-considered case that Q = x0; the only difference is that the dimension ofTr yν

∗Q ∩ T vtr y

is n− d rather than n. The crossing form evaluates with the same sign as f ′′(r)

on the radial tangent vector, and with the same sign as f ′(r) on each nonzero element of theorthogonal complement of the radial tangent vector. So we obtain

µRS(t 7→ (φ tH f)∗Tr yν

∗Q, T vtφ t

H f(r y)) =

n−d

2+Morse(π γ(r,y)) f ′(r)> 0, f ′′(r)> 0

n−d

2− 1+Morse(π γ(r,y)) f ′(r)> 0, f ′′(r)< 0

d−n

2+ 1−Morse(π γ(r,y)) f ′(r)< 0, f ′′(r)> 0

d−n

2−Morse(π γ(r,y)) f ′(r)< 0, f ′′(r)< 0

So by (16) we may generalize Proposition 3.1 as follows:

Proposition 3.3. For each element (r, y) ∈ Cc( f ,Q, x1), the corresponding element γ(r,y) ∈ C ri t(AH f

)

has Floer-theoretic grading given by

µ(γ(r,y)) =

d −Morse(π γ(r,y)) f ′(r)> 0, f ′′(r)> 0d + 1−Morse(π γ(r,y)) f ′(r)> 0, f ′′(r)< 0n− 1+Morse(π γ(r,y)) f ′(r)< 0, f ′′(r)> 0n+Morse(π γ(r,y)) f ′(r)< 0, f ′′(r)< 0

3.2. An assumption on the geodesics of N . Fix now a compact d-dimensional submanifoldQ ⊂ N and a point x1 ∈ N which neither lies in Q nor is a focal point of Q. Let

G (Q, x1) =

γ: [0, 1]→ N

γ(0) ∈Q, γ′(0) ∈ Tγ(0)Q

⊥, γ(1) = x1γ is a geodesic

Regarding any γ ∈ G (Q, x1) as a path in the zero section 0N ⊂ N , γ represents a class [γ] ∈π0(P (ν∗Q, T ∗

x1N)). Moreover γ is a critical point of the energy functional η 7→

∫|η′|2 on the

spacePN (Q, x1) of paths from P to x1, and this energy functional is Morse since we assume thatx1 is not a focal point of Q. Accordingly any γ ∈ G (Q, x1) has a Morse index, which as beforewe denote by Morse(γ). By [Mo34, Theorem V.15.2] this Morse index is equal to the number offocal points along γ, counted with multiplicity. For c ∈ π0(P (ν∗Q, T ∗

x1N)) and l ∈ Z we define

Gc,l(Q, x1) = γ ∈ G (Q, x1)|[γ] = c, Morse(γ) = l

For our main results we now make the following assumption on the behavior of geodesicsbetween Q and x1:

Assumption 3.4. There is a homotopy class c ∈ π0(P (ν∗Q, T ∗x1

N)) and an integer k such that:

(i) Gc,k(Q, x1) 6=∅.

(ii) Gc,k(Q, x1)∪Gc,k+2(Q, x1) is a finite set.

(iii) Gc,k−1(Q, x1)∪Gc,k+1(Q, x1) =∅

(iv) Either n− d 6= 2 or k 6= 0.

Remark 3.5. Since there are minimizing geodesics in every homotopy class of paths from Q tox1, and since such geodesics are always perpendicular to Q, it always holds that G

c,0(Q, x1) 6= ∅.In particular we can never have k = 1 in Assumption 3.4, as this would violate item (iii).

Remark 3.6. Assumption 3.4 is most often satisfied with k = 0; in fact, I do not know of anexample in which the condition holds for some nonzero k but does not hold for k = 0. Whenk = 0, (i) always holds (as mentioned in Remark 3.5), while (iii) requires only that G

c,1(Q, x1) =

∅.


Remark 3.7. Since the Morse complex generated by geodesics from Q to x1 has homology equalto that of space PN (Q, x1) of paths in N from Q to x1 (cf. the proof of Proposition 3.9 below),in order for Assumption 3.4 to hold it is necessary for P (Q, x1) to have a component withnontrivial homology in degree k but trivial homology in degrees k − 1 and k + 1. In the casethat Q is a single point, PN (Q, x1) is homeomorphic to the based loop space ΩN . In particular,if Q is a single point, then in order for Assumption 3.4 to hold with k = 0 it is necessary thatπ2(N) = 0 (since there are isomorphisms π2(N)

∼= π1(ΩN)∼= H1(ΩN ;Z)).

Remark 3.8. If the compact Riemannian manifold (N , g) has positive Ricci tensor everywhere,then Assumption 3.4 (ii) automatically holds. Indeed in this case [Mi63, Theorem 19.6] givesan upper bound on the length of any geodesic with index k or k+ 2, and since x1 is not a focalpoint of Q there can be only finitely many elements of G (Q, x1) whose lengths obey this upperbound.

Assumption 3.4 is frequently satisfied in nonpositive curvature:

Proposition 3.9. Let (N , g) be a compact connected Riemannian n-manifold and suppose that

there is c ≥ 0 such that all sectional curvatures of (N , g) are bounded above by −c. Let d < n with

d 6= n − 2 and suppose that Q ⊂ N is a compact d-dimensional submanifold such that for every

x ∈Q and every p ∈ T ∗xN with p|TQ = 0 and |p|= 1 the shape operator Σ(x ,p) : TxQ→ TxQ has all

of its eigenvalues bounded above byp

c. Then for any x1 ∈ N \Q and any c ∈ π0(P (ν∗Q, T ∗x1

N)),

Assumption 3.4 holds with respect to the data N , g,Q, x1, c, and k = 0.

Proof. By [W66, Theorem 4.1 and Corollary 4.2]2, the assumption on the curvature and on theeigenvalues of Σ(x ,p) imply that there are no focal points along any geodesic in G (Q, x1) (inparticular, x1 is not a focal point of Q). Consequently for any c we have G

c,l(Q, x1) = ∅ forl ≥ 1. Since we have assumed that d 6= n− 2 and since G

c,0(Q, x1) is nonempty, this proves allparts of Assumption 3.4 except for the statement that G

c,0(Q, x1) is finite. In fact we will showthat G

c,0(Q, x1) has just one element.3

The projectionπ: T ∗N → N induces a bijectionπ∗ betweenπ0(P (ν∗Q, T ∗x1

N)) and the spacePN (Q, x1) of paths from Q to x1, and just as in [Mi63, Theorem 17.3] the path component π∗cof PN (Q, x1) has the homotopy type of a cell complex with one l-cell for every element ofG

c,l(Q, x1) (see [K92, Section 3] for details on the extension of [Mi63, Theorem 17.3] to thecase where the left endpoints of the geodesics are replaced by a submanifold). But then sincewe have established that G

c,l(Q, x1) = ∅ for l ≥ 1, there are no cells of dimension greater thanone in this cell complex, and so since π∗c is path-connected there can only be one 0-cell. Thusindeed G

c,0(Q, x1) has just one element.

Meanwhile, here are some positive curvature cases in which Assumption 3.4 holds:

Proposition 3.10. Let (N , g) be either a compact semisimple Lie group with a bi-invariant metric,

or a sphere Sn where n ≥ 3 with its standard metric. Then Assumption 3.4 holds with k = 0 if we

choose Q to consist of a single point x0 which is not conjugate to x1.

2Note that [W66] uses an opposite sign convention to ours for the second fundamental form. Also, in the remarkabove [W66, Corollary 4.2], various references to c1/2 should be replaced by |c|1/2 throughout, as can be seen byconsidering the appropriate constant-curvature examples when c < 0.

3In the case that Q is totally geodesic, i.e. that each Σ(x ,p) = 0, one can show very directly that Gc(Q, x1) :=

∪lGc,l (Q, x1) has just one element (without appealing to [W66] or to the Morse index theorem) simply by observingthat if γ1,γ2 were distinct elements of G

c(Q, x1) then combining γ1 and γ2 with a suitable geodesic in Q from γ1(0)

to γ2(0) would yield a geodesic triangle in N having two right angles, which is impossible since N has nonpositivecurvature.

22 MICHAEL USHER

Proof. (N , g) has positive Ricci curvature (as is obvious in the case of Sn, and follows from [Mi63,p. 115] in the Lie group case since a semisimple Lie algebra has trivial center), so Assumption3.4(ii) holds by Remark 3.8. Of course Assumption 3.4(i) holds since we are taking k = 0, andAssumption 3.4(iv) holds since we have excluded S2 from the hypotheses and since there are notwo-dimensional compact semisimple Lie groups. As for Assumption 3.4(iii), [Mi63, Theorem21.7] is proven by showing that the geodesics connecting two nonconjugate points on a compactLie group always have even Morse index, so evidently G

c,1(x0, x1) =∅ in the Lie group case.Similarly, in the case of Sn all geodesics have index divisible by n−1, so that G

c,1(x0, x1) = ∅

(again using that n≥ 3).

Remark 3.11. There is a somewhat broader class of positively-curved symmetric spaces, whichincludes those from Proposition 3.10, to which Assumption 3.4 applies. Consider a compact-type symmetric space (N , g), given as a Riemannian quotient N = G/H where the compact Liegroup G is the identity component of the isometry group of N and the isotropy group H is aunion of components of the fixed locus of an involution σ of G. Decompose the Lie algebra g ofG as g = h⊕ p where h is the Lie algebra of H and p is the (−1)-eigenspace of the linearizationof σ. Thus the tangent space Tx0

N at a suitable basepoint x0 ∈ N is naturally identified with p,and we have

[h,h] ⊂ h, [p,h] ⊂ p, [p,p] ⊂ h.

Under the identification Tx0N ∼= p the curvature tensor of (N , g) at x0 is given by

R(X , Y )Z = −[[X , Y ], Z] (X , Y, Z ∈ p),and moreover the curvature tensor is parallel (see, e.g., [He78, Chapter IV]). So if γ is a geodesicwith γ′(0) = X ∈ Tx0

N , then t is a conjugate time for γ if and only if t =2π jpλ

where j ∈ Z+and λ is a positive eigenvalue of the operator Y 7→ [[X , Y ], X ] on p, with the multiplicity of theconjugate time equal to the multiplicity of the corresponding eigenvalue λ.

In particular, it follows that N = G/H obeys Assumption 3.4 with k = 0, with x0 equal to anypoint not conjugate to x1, and with any homotopy class c, provided that the following conditionholds:(17)For all X ∈ p, the largest eigenvalue of (Y ∈ p) 7→ [[X , Y ], X ] has multiplicity greater than one

(Indeed, the argument just given shows that, Gc,1(x0, x1) = ∅, and since N is assumed to be

of compact type it has positive Ricci tensor, so Assumption 3.4 (ii) holds by Remark 3.8. Also,since [[X , X ], X ] = 0 the condition (17) forces dim N ≥ 3, so Assumption 3.4 (iv) holds.)

To verify (17) in a given example, it is convenient to note that it only needs to be checkedfor all X belonging to a fixed maximal abelian subspace a ⊂ p, since [He78, Lemma V.6.3]shows that there is an isometry of N whose derivative maps any given element of p into a. Withthis said, we leave it to the reader to check that (17) and hence Assumption 3.4 holds for thefollowing classes of symmetric spaces (for all but the first and last of these, the subspace p ⊂ g

and a convenient choice of the maximal abelian subspace a ⊂ p are indicated in [He78, SectionX.2.3]; for the case of rank-one symmetric spaces, including OP2, see also [He65, p. 171]—therelevant eigenvalue multiplicity is denoted there by q):

• G0∼= G0×G0

∆where G0 is a compact semisimple Lie group and∆≤ G0×G0 is the diagonal.

• The spheres Sn =SO(n+1)

SO(n)and real projective spaces RPn =

SO(n+1)S(O(1)×O(n))

, where n≥ 3.

• The quaternionic Grassmannians Grm(Hn) =

Sp(n)

Sp(m)×Sp(n−m)for 1≤ m≤ n− 1.

• The space U(2n)

Sp(n)of unitary quaternionic structures on C2n.


• The Cayley projective plane OP2 =F4

Spin(9).

On the other hand, the members of those infinite families in Cartan’s classification of irre-ducible compact-type symmetric spaces that are not listed above all have nonzero π2 and so, byRemark 3.7, cannot satisfy Assumption 3.4 with k = 0 and Q equal to a singleton.

Note also that if (N , g) is instead just locally isometric to a compact-type symmetric space G/Hobeying (17) then the same argument applies to show that Assumption 3.4 holds for (N , g). (Forinstance, N could be a quotient of G/H by a discrete group of isometries.)

Remark 3.12. If the manifold (N , g) (together with auxiliary data Q, x1, c) satisfies Assumption3.4, and if (N ′, g ′) is an arbitrary compact Riemannian manifold, then the product (N×N ′, g×g ′)also satisfies Assumption 3.4 (using auxiliary data Q×N ′, (x1, x2), c×c0 where x2 ∈ N ′ is chosenarbitrarily and c0 is the homotopy class of the constant path at x2). Indeed, it is easy to see thatthe geodesics in the product N × N ′ from Q × N ′ to (x1, x2) which represent the class c × c0and are initially orthogonal to Q × N ′ are precisely maps of the form η: t 7→ (η(t), x2) whereη is a geodesic in N from Q to x1 which is initially orthogonal to Q. Moreover, the homotopyclasses c× c0 in π0(P (Q × N ′, (x1, x2))) remain distinct as c varies through π0(P (Q, x1)), andthe conjugate times and multiplicities for a geodesic η from Q to x1 are the same as those for itscorresponding geodesic η, so that the Morse indices coincide under the correspondence η↔ η.

In particular we obtain in this way Riemannian manifolds obeying Assumption 3.4 that haveindefinite Ricci tensor, to go along with our previous positively- and negatively-curved examples.

4. ESTIMATING THE BOUNDARY DEPTH

Let (N , g) be a compact connected Riemannian n-manifold satisfying Assumption 3.4 withrespect to a compact d-submanifold Q and a point x1, a nonnegative integer k, and a homotopyclass c. We consider the Floer complexes C F

c(ν∗Q, T ∗

x1N ; H f

a

) for certain functions f~a : [0,∞)→R associated to vectors ~a ∈ R∞ that will be described presently.

First let us fix an arbitrary R > 0 and a smooth function h: R → [0, 1] with the followingproperties:

• supp(h) = [δ, R−δ] for some real number δ with 0< δ < R

4.

• The only local extremum of h|(δ,R−δ) is a maximum, at h(R/2) = 1.• h′′(s)< 0 if and only if s ∈ (R/4, 3R/4).• h′′(s)> 0 if and only if s ∈ (δ, R/4)∪ (3R/4, R− δ).

Now for ~a ∈ R∞ define

f~a(s) =

∞∑

i=0

ai f (2i+1s− R)

So the restriction of f~a to each interval [2−(i+1)R, 2−iR] is a copy of h which has been rescaledhorizontally to have support within the interval and which has been rescaled vertically by thefactor ai .

Proposition 4.1. Under Assumption 3.4, there is a constant C0 > 0 depending on the Riemannian

metric g and the function h but not on ~a with the properties that, whenever ν∗Q ⋔ (φ1H f~a

)−1(T ∗x1

N):

(i) For every (r, y) ∈ Cc( f~a,Q, x1) such that µ(γ(r,y)) ∈ d − k − 1, d − k + 1 we have

AH f~a(γ(r,y))≥ −C0.

(ii) For every (r, y) ∈ Cc( f~a,Q, x1) such that µ(γ(r,y)) ∈ n + k − 1, n + k + 1 we have

AH f~a(γ(r,y))≤ C0.

24 MICHAEL USHER

Proof. We first prove (i). Consulting Proposition 3.3, we see that any such (r, y) would corre-spond to a geodesic π γ(r,y) whose Morse index is one of the following:

(A) k± 1, if f ′~a(r)> 0, f ′′

~a(r)> 0

(B) k or k+ 2, if f ′~a(r)> 0, f ′′

~a(r)< 0

(C) −k− (n− d) or 2− k− (n− d), if f ′~a(r)< 0, f ′′

~a(r)> 0

(D) −(n− d)− k± 1, if f ′~a(r)< 0, f ′′

~a(r)< 0

Now (A) above is forbidden by Assumption 3.4(iii). (C) is also forbidden: for the first subcaseof (C) just note that there are no geodesics of negative Morse index and we have k ≥ 0 andn − d ≥ 1 (as Q is a compact d-submanifold of the connected n-manifold N which does notcontain x1 and so has positive codimension), so the Morse index cannot be −k − (n − d). Ifthe Morse index were 2 − k − (n − d), then in view of Assumption 3.4(iv) and the facts thatn − d ≥ 1 and (by Remark 3.5) k 6= 1, it would have to hold that k = 0 and n − d = 1. Butthis is also impossible, since then Morse(π γ(r,y)) would be 1 whereas by Assumption 3.4(iii)G

c,1(Q, x1) = ∅. So one of (B) or (D) applies, and in particular our element (r, y) ∈ Cc( f~a,Q, x1)

has f ′′~a(r) < 0. Also, the only way that (D) could hold is if k = 0, n− d = 1, and π γ(r,y) has

index zero. So in any case π γ(r,y) has index either k or k+ 2.In view of Assumption 3.4(ii) there is an upper bound, say L, on the lengths of index-k or

k+ 2 geodesics in class c from Q to x1. Our element (r, y) ∈ Cc( f~a,Q, x1) has the property that

± f ′~a(r) is equal to the length of such a geodesic. NowAH f~a

(γ(r,y)) = f~a(r)− r f ′~a(r), and we have

|r f ′~a(r)| ≤ RL

So if f~a(r) ≥ 0 obviously AH f~a(γ(r,y)) ≥ −RL. So assume f~a(r) < 0. Choosing j such that r ∈

[2−( j+1)R, 2− j r], we have f~a(r) = a jh(2j+1r−R) (so a j < 0). Meanwhile f ′~a(r) = 2 j+1a jh

′(2 j+1r−R) has absolute value equal to the length of an index-k or k+2 geodesic from Q to x1, and henceis bounded above by L.

By construction, h′′ is positive precisely on the intervals (δ, R/4) and (3R/4, R−δ). So sincea j < 0 (so f ′′~a (r), which was earlier shown to be negative, has opposite sign to h′′(2 j+1r − R)),we have 2 j+1r−R ∈ (δ, R/4)∪ (3R/4, R−δ). If 2 j+1r−R ∈ (δ, R/4), then since a j < 0, h(δ) = 0and h′ is nonnegative and increasing on [δ, 2 j+1r − R] we see that

f~a(r) = a jh(2j+1r − R) = a j

∫ 2 j+1 r−R

δ

h′(s)ds ≥ Ra jh′(2 j+1r − R)≥ −RL

Similarly if 2 j+1r−R ∈ (3R/4, R−δ) then since h(R−δ) = 0 and h′ is nonpositive and increasingon [2 j+1r − R, R− δ],

f~a(r) = a jh(2j+1r − R) = −a j

∫ R−δ

2 j+1 r−R

h′(s)ds ≥ −Ra jh′(2 j+1r − R)≥ −RL

Summing up, the relevant γ(r,y) have AH f~a(γ(r,y)) = f~a(r)− r f ′~a(r) where both f~a(r) ≥ −RL

and r f ′~a(r)≥ −RL, so part (i) of the Proposition holds with C0 = 2RL.The proof of part (ii) is a mirror image to that of part (i): one uses Proposition 3.3 to see

that any such (r, y) must have f ′′~a(r) > 0 and must correspond to a geodesic of Morse index k

or k+ 2. This fact leads to an estimate f~a(r)≤ RL, and hence to

AH f~a(γ(r,y)) = f~a(r)− r f ′~a(r)≤ 2RL

Details are left to the reader.


Proposition 4.2. Under Assumption 3.4 there is a constant A> 0 depending on (N , g) and h such

that for all ~a ∈ R∞ obeying ν∗Q ⋔ (φ1H f~a

)−1(T ∗x1

N):

(i) If mini ai < −A then there exists (r, y) ∈ Cc( f~a,Q, x1) such that µ(γ(r,y)) = d − k and

AH f~a(γ(r,y))≤mini ai .

(ii) If maxi ai > A then there exists (r, y) ∈ Cc( f~a,Q, x1) such that µ(γ(r,y)) = n + k and

AH f~a(γ(r,y))≥maxi ai .

Proof. Let lk denote the minimal length of an index-k geodesic from Q to x1 in class c (suchexists by Assumption 3.4(i)). Let A= − lk

2h′(3R/4)(note that it follows from the construction of h

that h′(s) attains its minimal value, which is negative, at s = 3R/4). Thus if b > 2A, then thereis s ∈ (R/2, 3R/4) such that bh′(s) = −lk.

If mini ai < −A, choose j so that a j =mini ai . For each r ∈ [2−( j+1)R, 2− jR] we have f ′~a(r) =2 j+1a jh

′(2 j+1r − R). So since −2 j+1a j > 2A there is r0 such that 2 j+1r0 − R ∈ (R/2, 3R/4) andf ′~a(r0) = lk. Since a j < 0 and, by the construction of h, h′′ is negative on (R/2, 3R/4), we have

f ′′~a(r0)> 0.Since f ′

~a(r0) = lk is the length of a Morse index-k geodesic from x0 to x1 in the class c, there

is a corresponding element (r0, y) ∈ Cc( f~a,Q, x1). Since f ′

~a(r0) and f ′′

~a(r0) are both positive we

have µ(γ(r0,y)) = d−k by Proposition 3.3. As for the action, let r1 =342− jR, so that 2 j+1r1−R= R

2and so f~a(r1) = a j , f ′

~a(r1) = 0 and f ′′

~ais positive on [r1, r0]. Hence

AH f~a(γ(r0,y)) = f~a(r0)− r0 f ′~a(r0)

= f~a(r1) +

∫ r0

r1

d

dr( f~a(r)− r f ′~a(r))dr

= a j −∫ r0

r1

r f ′′~a (r)dr ≤ a j =mini

ai

This proves (i)The proof of (ii) is essentially the same: choose j so that a j =maxi ai > A, and then there is r0

with 2 j+1r0−R ∈ (R/2, 3R/4) such that f ′~a(r0) = 2 j+1a j g

′(2 j+1r0−R) = −lk. Moreover there is acorresponding (r0, y) ∈ C

c( f~a,Q, x1), which will have µ(γ(r0,y)) = n+k since f ′

~a(r0), f ′′

~a(r0)< 0,

and will haveAH f~a(γ(r0,y))≥ a j by a similar calculation to that above.

This quickly leads to the key estimate of the boundary depth B as defined in (10).

Proposition 4.3. Under Assumption 3.4 there is a constant C such that, for all ~a ∈ R∞,

(18) B

ν∗Q, (φ1

H f~a

)−1(T ∗x1

N)

≥ ‖~a‖∞ − C

Proof. By construction, each of the functions f ′~a

have only finitely many critical values. From this(and the fact that x1 is not a focal point of Q, so that there are only countably many geodesicsfrom Q to x1) it is easy to see that, for any ~b ∈ R∞, all but countably many λ ∈ R have theproperty that the transversality condition (12) holds for the function H fλ~b

= Hλ f~b. So using the

continuity property (8) of the boundary depth it suffices to prove the proposition for those ~a ∈R∞ such that ν∗Q ⋔ (φ1

H f~a

)−1(T ∗x1

N) (so that in particular the Floer complex C F(ν∗Q, T ∗x1

N ; H f~a)

is well-defined).

26 MICHAEL USHER

Let C be the maximum of the constant C0 from Proposition 4.1 and the constant A fromProposition 4.2. Then (18) is trivial if ‖~a‖∞ ≤ C (since B is by definition always nonnegative),so we may assume that ‖~a‖∞ > C .

Now either ‖~a‖∞ = −mini ai or ‖~a‖∞ = maxi ai . In the first case, we have mini ai < −C ≤−A, So by Proposition 4.2(i) there is a generator γ(r,y) for C F

c,d−k(ν∗Q, T ∗

x1N ; H f~a

) with filtrationlevel at most−‖~a‖∞. But by Proposition 4.1(i), there are no generators of C F

c,d−k−1(ν∗Q, T ∗

x1N ; H f~a

)

having action less than or equal to −C . This implies first that ∂Jγ(r,y) = 0 (where ∂J is the Floerdifferential), since ∂J maps C F

c,d−k to C Fc,d−k−1 and lowers filtration level, and the filtration

level of γ(r,y) is at most−‖~a‖∞ < −C . But the homology of the chain complex (C Fc,d−k(ν

∗Q, T ∗x1

N ; H f~a),∂J)

is zero (it is independent of the choice of compactly supported Hamiltonian H, and clearly van-ishes when H = 0 since ν∗Q ∩ T ∗

x1N = ∅). So the fact that ∂Jγ(r,y) = 0 implies that

γ(r,y) ∈ Im∂J : C F

c,d−k+1(ν∗Q, T ∗

x1N ; H f~a

)→ C Fc,d−k(ν

∗Q, T ∗x1

N ; H f~a)

By another application of Proposition 4.1(i), all nonzero elements of C Fc,d−k+1(ν

∗Q, T ∗x1

N ; H f~a)

have filtration level at least −C . Thus γ(r,y) is an element of the image of the boundary operatorhaving filtration level at most −‖~a‖∞, all of whose primitives have filtration level at least −C .This proves (18) in the case that ‖~a‖∞ = −mini ai .

There remains the case that ‖~a‖∞ = maxi ai . This case becomes essentially identical to theother one after we appeal to [U10, Corollary 1.4], which shows that the boundary depth ofthe Floer complex C F

c(ν∗Q, T ∗

x1N ; H f~a

) is the same as that of its “opposite complex,” i.e. of thefiltered chain complex obtained from C F

c(ν∗Q, T ∗

x1N ; H f~a

) by negating the grading, using theopposite filtration

ℓ∑

biγi

=max−AH f~a

(γi)|bi 6= 0,and using as boundary operator the transpose of the original boundary operator ∂J. In thisopposite complex, the generator γ(r,y) provided by Proposition 4.2(ii) will have grading −n− k

and filtration level at most −maxi ai = −‖~a‖∞, while Proposition 4.1(ii) shows that γ(r,y) is acycle (and hence a boundary) in the opposite complex (since all index-(−n− k− 1) generatorshave higher filtration level), and that all primitives of γ(r,y) have filtration level at least −C .Thus in the case that ‖~a‖∞ =maxi ai we again obtain (18).

The k = 0 case of the following corollary immediately implies Theorem 1.2:

Corollary 4.4. Let the Riemannian manifold (N , g), the submanifold Q, and the point x1 ∈ N \Qwhich is not a focal point of Q obey Assumption 3.4. Let U be a neighborhood of 0N , choose

R > 0 sufficiently small that the radius-R disk bundle in T ∗N is contained in U, and construct the

functions f~a as at the start of Section 4. Then the homomorphism F : R∞→ C∞(T ∗N) defined by

F(~a) = H f~asatisfies the following properties, for all ~a,~b ∈ R∞:

(i) φ1F(~a+~b)

= φ1F(~a)φ1

F(~b)

(ii) For some constant C independent of ~a and ~b,

δφ1

F(~a)(T∗x1

N),φ1F(~b)(T ∗

x1N)

≥ ‖~a− ~b‖∞ − C

and

δφ1

F(~a)(ν∗Q),φ1

F(~b)(ν∗Q)≥ ‖~a− ~b‖∞ − C

where δ denotes the Hofer distance on L (T ∗x1

N) or on L (ν∗Q), respectively.


Proof. It is immediate from the definition of f~a that f~a+~b = f~a + f~b; thus F is indeed a homo-morphism and the formula (11) for the Hamiltonian flows of the functions H f directly implies(i). As for (ii), note that for any Lagrangian submanifold Λ, the Ham-invariance of the Hoferdistance δ, together with (i), shows that we have

δ(φ1F(~a)(Λ),φ

1F(~b)(Λ)) = δ(Λ, (φ1

F(~a−~b))−1(Λ)).

Now B(ν∗Q, T ∗x1

N) = 0 since the Floer complex C F(ν∗Q, T ∗x1

N ; 0) vanishes. Hence by Corol-laries 2.4 and 4.3,

δ(T ∗x1

N , (φ1F(~a−~b))

−1(T ∗x1

N))≥ B(ν∗Q, (φ1F(~a−~b))

−1(T ∗x1

N))≥ ‖~a− ~b‖∞ − C ,

proving the first line of (ii). As for the second line of (ii), note that for all ~a the Floer complexC F(T ∗

x1Q,ν∗Q; F(~a)) is naturally isomorphic (modulo a shift of grading that results from our nor-

malization convention) as a filtered chain complex to the opposite complex of C F(ν∗Q, T ∗x1

N ; F(−~a))(in the sense of [U10] and of the last paragraph of Proposition 4.3). So since by [U10, Corollary1.4] the boundary depth is unchanged when we pass to the opposite complex, we obtain

δ(ν∗Q, (φ1F(~a−~b))

−1(ν∗Q))≥B(T ∗x1

N , (φ1F(~a−~b))

−1(ν∗Q))− B(T ∗x1

N ,ν∗Q)

= B

T ∗

x1N , (φ1

F(~a−~b))−1(ν∗Q)≥ ‖~a− ~b‖∞ − C .

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28 MICHAEL USHER

[RS93] J. Robbin and D. Salamon. The Maslov index for paths. Topology 32 (1993), no. 4, 827-844. MR1241874.[U10] M. Usher. Duality in filtered Floer-Novikov complexes. J. Topol. Anal. 2 (2010), no. 2, 233-258. MR2652908.[U11a] M. Usher. Boundary depth in Hamiltonian Floer theory and its applications to Hamiltonian dynamics and

coisotropic submanifolds. Israel J. Math. 184 (2011), 1-57. MR2823968.[U11b] M. Usher. Hofer’s metrics and boundary depth. arXiv:1107.4599, to appear in Ann. Sci. Éc. Norm. Sup.[U12] M. Usher. Submanifolds and the Hofer norm. arXiv:1201.2926, to appear in J. Eur. Math. Soc.[W66] F. Warner. Extensions of the Rauch comparison theorem to submanifolds. Trans. Amer. Math. Soc. 122 (1966),

341-356. MR0200873.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF GEORGIA, ATHENS, GA 30602E-mail address:

Documents

HOFER GEOMETRY AND COTANGENT FIBERSmath.uga.edu/~usher/cotangent.pdf · HOFER GEOMETRY AND COTANGENT FIBERS 3 denote the orbit of S under the Hamiltonian diffeomorphism group. The